Model predictive control for scalar linear systems with cauchy noises

A model predictive optimal control scheme is developed for scalar discrete linear dynamic systems with Cauchy distributed process and measurement noises. The second moment cost criterion, commonly used for systems with Gaussian noises, cannot be used to address the control problem with Cauchy noises since their probability density functions have an undefined first moment and an infinite second moment, leading to an infinite second-moment cost. Therefore, a cost criterion, functionally resembling the Cauchy density, is chosen. It is shown that the unconditional expectation of this criterion with respect to the Cauchy densities exists. A model predictive control scheme that at each time step minimizes the unconditional expected cost to some fixed time horizon is then derived. Numerical results are shown for different horizon lengths. The Cauchy controller is compared to a controller that assumes Gaussian noises. An essential difference between the two is that, as opposed to the Gaussian controller, the Cauchy controller does not produce large control signals in response to large measurement noises. Conversely, impulsive process noises that drive the system induce larger control response needed for regulation while using the Cauchy controller.